Timedependent approach to semiclassical dynamics Eric J. Heller Citation: J. Chem. Phys. 62, 1544 (1975); doi: 10.1063/1.430620 View online: http://dx.doi.org/10.1063/1.430620 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v62/i4 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 18 Oct 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
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Timedependent approach to semiclassical dynamicsEric J. Heller Citation: J. Chem. Phys. 62, 1544 (1975); doi: 10.1063/1.430620 View online: http://dx.doi.org/10.1063/1.430620 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v62/i4 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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Time-dependent approach to semiclassical dynamics Eric J. Heller
The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (Received 9 September 1974)
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time-dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many-particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton's equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State-specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H, system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
I. INTRODUCTION
In recent years much attention has been directed towards the investigation of the semiclassical limit of quantum mechanics. To a large extent, these recent efforts have been motivated by the fact that many aspects of molecular quantum dynamics fall into the semiclassical domain. In fact, for many systems it is only because we ask quantum questions (such as energy specific transition probabilities between two quantum states) that we need to invoke quantum mechanics at all. As Miller l has emphasized, suitable averages very quickly tend to wash out quantum effects. Nevertheless, with increasingly sophisticated excitation and detection techniques becoming available, it is quite possible to measure quantum state-specific properties of even very "heavy" molecular systems. For the same systems, quantitative quantum mechanical calculations may become prohibitive because of the large number of quantum states which must be coupled to describe eigenstates for heavy particles.
Current trends in the application of semiclassical techniques to molecular dynamics can be traced back to the correspo,ndence principle and the early views of Dirac:>' through the Feynman path integral approach3 and up to the more recent work of Miller, 4 Marcus, 5 Gutzwiller, 6 and many other important contributions. There exists by now a rather large literature on semiclassical dynamics employing the principle of expansion in 11 (see Dirac's book,:>' and in this context the paper by Eu') and the correspondence between stationary phase approximations to the quantum mechanical propagators and classical paths, as emphasized by Miller.
In this paper we examine an alternative avenue of approach to the semiclassical limit of quantum mechanics. The approximations involved in the theory presented below differ significantly from those of the semiclassical techniques mentioned above. It is hoped that the present theory will provide an alternative framework which may be valid in certain regimes which are complementary to existing techniques.
Most of classical mechanics deals with the time-dependent equations of motion for a system of point particles. The closest semiclassical analog to this is a corresponding system of localized quantum wave packets. The intuitive appeal of a wave packet picture is evident, but a practical scheme needs to be developed in order to exploit the wave packet picture for the purpose of obtaining specific quantum information. It is the purpose of this paper to provide such a scheme. We deal here with a single form for the wave packets, namely multivariate Gaussians. A succeeding paper will consider more general forms for semiclassical wavefunctions and a variational approach to the time dependence of these wavefunctions, allowing treatment of classes of problems (such as curve crOSSing) which cannot be handled with the technique presented here.
In a section of his book entitled "The Motion of Wave Packets, " Dirac2 has succinctly stated the point of departure for our investigation. We quote from Dirac: "For any dynamical system with a classical analogue, a state for which the classical description is valid as an approximation is represented in quantum mechanics by a wave packet, all the coordinates and momenta having approximate numerical values ... Schrooinger's wave equation fixes how such a wave packet varies with time, so in order that the classical description may remain valid, the wave packet should remain a wave packet and should move according to the laws of classical dynamics. "
Dirac then introduces the form l/J=Aexp[(i/If)S] and proceeds to derive equations of motion for A and S, on the assumption that If is small. It is here that we adopt a different viewpOint, noting that for harmonic potentials, the above quote from Dirac is valid no matter how big or small If is. In harmonic potentials, Gaussian wave packets remain Gaussian, and undergo periodic motion, and the expectation values of positions and momenta undergo the classical equations of motion. In fact, we are just stating a result of Ehrenfest's theorem, 8 which tells us, for example,
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Eric J. Heller: Time-dependent approach to semiclassical dynamics 1545
.!£ (p) = _ aH«p), (q» +error term dt a(q) (1.1)
error term-d2(a3V/aq3)/(aV/aq)!.=<.> ,
where H= Ho + V in the Hamiltonian for the system, (p), (q) are numbers representing the expectation values of the corresponding quantum operators over the wavefunction at time t, and d is the diameter of the packet. The points we are now emphasizing are that the error term is independent of n and vanishes if a3v/aq3 = O. We also point out the fact that the error is small if d is small. 9
This helps to motivate the approximation which we wish to introduce, in order to treat nonharmonic potentials without recourse to stationary phase and expansions in n. We assert that, for sufficiently narrow wave packets and relatively smooth potentials, the wave packet is so restricted in coordinate space that it doesn't "feel" more than the first few terms in the Taylor expansion of the potential about the center of the wave packet. If we retain terms up to and including (x - Xt)2 in the expansion of the potential (where Xt is the center of the wave packet), then at each instant of time the wave packet feels a harmonic potential, so initially Gaussian wavefunctions must remain Gaussian. This leads us to consider a wavefunction of the form
which is a general time dependent Gaussian wave packet. Thus, we postulate a convenient form for IjJ and determine equations of motion for its parameters. However, instead of expanding in terms of n, we approximate the potential locally. We specifically wish to avoid any expansion in n, since this would ultimately lead us back to the generalized WKB, classical S-matrix version of semiclassical dynamics. We are attempting to develop a procedure which rests on somewhat different assumptions than the classical S-matrix theory imposes. Accordingly, our view of the classical correspondence of quantum dynamics is not strictly the usual n - 0 limit. Rather, we exploit the correspondence between classical dynamics and the quantum dynamics of localized wave packets on smooth potential surfaces.
In Sec. II we derive equations of motion for the Gaussian parameters under our assumption on the expansion of the potential about the instantaneous center of the wave packet. In Sec. III, we show how nonseparable systems may be treated and how interparticle correlations can be included in the Gaussian wavefunction. Section IV discusses various techniques for obtaining the S matrix, and Sec. V contains a numerical application to the He + H2 collinear inelastic system. Section VI discusses possible generalizations and future work. Finally, Sec. VII contains a brief conclusion and a discussion of related work dealing with wave packets.
II. ONE-DIMENSIONAL SYSTEMS
A. Quadratic potentials
As noted in the Introduction, wave packets which are initially Gaussian remain Gaussian in quadratic poten-
tials. Q,tadratic potentials will be taken to include harmoniC, linear, and constant functions of the position variable x. The exact quantum mechanical propagator (space time Green's function) is known for such potentials, 10 and it is easy to show by application of these propagators to Gaussian wave packets that the wave packets do indeed remain Gaussian, although they may spread, acquire phase factors, etc.
One of the advantages of a Gaussian function is that a very few parameters serve to specify a complete quantum wavefunction. Let us rewrite Eq. (1. 2) in a more suggestive form:
The parameters at, Yt are complex numbers and Xt , Pt are taken to be real. If Yt has an imaginary part such that (1jJ 11jJ) = 1, then it is easily verified that
(p) (;;; (ljJ!p!IjJ»=Pt ,
and (2.2)
so these parameters have their classical meaning.
There are four parameters in Eq. (2.1), two complex and two real. In classical mechaniCS, only xt and Pt are needed to specify the motion of the system. The classical equations of motion are
dpt _ aH I -a:t-- ax "="t'
dXt _ aHI -a:t- - ap P=.t •
(2.3)
We shall try to mimic classical mechanics as closely as possible by seeking first order equations of motion for the parameters in Eq. (2.1). This is easily accomplished by inserting Eq. (2.1) into the SchrOdinger equation. The Hamiltonian governing the motion is
(2.4)
where Vex) is, for now, at most a quadratic function of x. We rewrite Has
etc. The parameter xt appearing in Eq. (2.5) is just the "center of the wave packet" appearing in Eq. (2.1). No approximation to H has been made in Eq. (2.5) since the Taylor series expansion is exact for the potentials now under consideration. Inserting (2.1) and (2.5) into
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1546 Eric J. Heller: Time-dependent approach to semiclassical dynamics
(2.7)
where at = dati dt, etc. Comparing coefficients of like powers of (x - x t ), we find that Eq. (2.7) is satisfied if
at = - (2/m)a~ - Vu /2 ,
2atxt - Pt = 2atp/ m + Vx ,
Yt = inat/m +PtXt - Vo - p~/2m .
Equation (2.8b) makes sense if we define
Xt = Pt/m = aH/ap Ip=pt .
Noting that Vx = aH/ax Ix=xt and that
H(Pt, Xt) = pV2m + V(xt ) = E ,
our equations of motion become
Xt =aH/aPt ,
Pt = - aH/axt ,
at = - (2/m)a~ - %Vu ,
Yt=inut/m+Ptxt- E .
(2.8a)
(2.8b)
(2.8c)
(2.9)
(2.10)
(2.11a)
(2.11b)
(2.11c)
(2.11d)
No approximations have been introduced, so that if Eqs. (2.11) are solved, Eq. (2.1) gives the exact result. For V(X) = 0, we have
Pt =Po ,
(2. 12a)
(2. 12b)
a ao (2. 12c) t (2/ m)ao(t - to) + 1 '
- in l 120'01 iIl l (20'0(t-to) 1) E(t-t) Yt - - 4 n " + 2 n + + 0 • 1[" m (2. 12d)
Similarly, for V(x) = %mw2~, Xt and Pt execute the usual harmonic motion while
a ,,_~(%mw - aocot(wt») t 2 ao+%mwcot(wt)
(2.13)
Note that if ao = imw/2n, we have at = ao for all t. The Gaussian wave packet in this case is called a Glauber coherent state, 11 having useful properties which we shall employ later on.
B. Locally quadratic potentials
It is not a very large step from the results of the last section, which were exact solutions for simple potentials, to approximate solutions for more difficult potentials. As long as the potential is smoothly varying over the width of the wave packet, it is reasonable to assume that for the purpose of propagating the solution to time t + 0 t a quadratic expansion of V(x) about xt is valid. Some error is inevitably incurred unless the potential is quadratic, but this error should be small for welllocalized wave packets and/or smooth potentials. These are, of course, familiar semiclassical criteria.
Thus we propose to make the substitution, Eq. (2.5), in place of Eq. (2.4), for arbitrary V(x). This makes the new, effective Hamiltonian time dependent, but the difference between the exact and effective Hamiltonian is smallest where the wavefunction is largest.
The derivations leading to Eqs. (2.11) are unchanged. For arbitary V(x) we must integrate the first order classical and quantum trajectories, Eqs. (2.11), numerically. This is easily accomplished with standard techniques.
It is important to note that the potential is expanded quadratically about the instantaneous center of the wave packet. We do not at this point envision propagating the classical parameters Xt , Pt on an approximate harmonic potential surface expanded about several fixed points. This might be a reasonable additional approximation for future examination. For now, however, xt and Pt follow the exact classical motion on a smooth potential surface, while at and Yt experience the effects of smoothly time varying linear forces.
The Fourier transform p-space counterpart of the wavefunction given by Eq. (2.1) is
1 r~ Iji(p)=rz:;; L dxexp(- ipx)lji(x)
= (- 2iO't f l /2 exp( - i1f(p - Pt )2/4at - ipx/n + iYt/n) , (2.14)
which is a Gaussian centered about the classical momentum Pt. Thus, our wave packet approach may be interpreted as consisting of a point in phase space, representing the center of the wave packet, executing a classical trajectory; this point is "clothed" by a Gaussian function containing the information about the quantum part of the problem. The Gaussian is able to penetrate into classically forbidden territory. This property allows for a correct description of classical turning pOints; there are no singularity difficulties at such points. Furthermore, as we shall see in the numerical application of Sec. V, certain other classically forbidden processes, such as the dynamically inaccessible n- n' transitions in atom-diatom collinear collisions are correctly accounted for with no additional assumptions. Paradoxically, direct barrier tunneling will evidently require further work, since the center of the Gaussian wave packet always follows a completely classical trajectory in the present formulation.
C. Phase of the wave packet, eigenfunctions
The Gaussian wavefunction Eq. (2.1), at its center X" xt , is simply
(2.14')
Examining the three terms in the expression (2. 11d) for Yt, we see that the first gives rise to a purely logarithmic term in the case V = 0 [Eq. (2. 12d)] and thus goes "downstairs" as a time-dependent normalization factor in Iji. On the other hand, for the coherent state of the harmonic OSCillator, we have at = imw/2, and thus the term involving at in Eq. (2. 11d) becomes - fV...J/2, which becomes a pure phase factor responsible for adding the correct zero point quantum energy of 1/2nw onto the classical energy E appearing in Eq. (2. 11d). (For example, the wave packet representing the ground vibrational state is at rest; i. e., E = 0 in Eq. (2. l1d). ]
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Eric J. Heller: Time-dependent approach to semiclassical dynamics 1547
The remaining terms in Yt, namely
<P ;: f (Pt' x" - E)dt'
= r [p~,/2m - V(xt , )]dt'
;: r £dt' (2.15)
are recognized as the action integral along the trajectory.12 Thus, in the course of its trajectory, the wave packet acquires a phase factor
e(iMq, = exp G/n rt p(x)dx- iEt/n] , (2.16)
Xo
where p(x) is the classical momentum at x for total energy E.
Suppose now that we send a wave packet from large positive XI toward X = 0, allow it to "scatter" from V(x), and wait until it again returns to large x" We may extract an energy eigenfunction from such a wave packet trajectory Simply by a Fourier transform of I/J(x, t):
(2. 17)
Figure 1 (a) shows the wave packet at its turning pOint. It is possible to choose the initial parameters such that the turning point occurs at t= 0 and that at=o is purely imaginary. Then we will have
as can easily be seen from the symmetry of the classical motion about the turning point and Eqs. (2.11). Equations (2.18) have as a consequence
1/J(x, t) = tfJ*(x, - t) (2.19)
and, as a result, Eq. (2.17) is seen to yield a purely real function of x. This is plotted schematically in Fig. l(b). Figure 1 shows that both the wave packet and its Fourier transform behave properly at the turning point. Writing Eq. (2.17) explicitly, we have
x exp [i/n f Pt"Xt"dt" - f' at;:t" + iYoIn]. (2.20)
Since for 0 ~ t!(, 00 there is a unique relationship between t ' and xt ., we may take x" to be the independent variable writing
dt ' = dxt• (dt' / dxt·) ,
Pt' ;: p(xt·) ,
1/' f."I' i/n Pt";;I"dt" = i/n p(xt• )dXt' o Xo
f."t.
;: i/n p(x)dx . xo
The phase factor
s."t'
i/n p(x)dx Xo
(2.21)
appearing in Eqs. (2.21) and thus in Eq. (2.20) is reminiscent of a similar term in the WKB wavefunction. We may see the relationship with the WKB phase more expliCitly by assuming that our time dependent wave packet 1/J(x, t) is propagating in free space after having interacted with V(x). The exact wavefunction which started out Gaussian and scattered off V(x) is expressible in the form
with A(P) given by the p-space amplitude for the Gaussian [see Eq. (2.14)]. To find the phase shift at momentum Po, we project 1J!~~f't onto exp[ - ipox/n]:
An estimate of o(Po) can apparently be had by substituting our apprOXimate outgoing wave packet into Eq. (2.23). The resulting expression for o(Po) is
o(Po) = - poxoln +~ 100
dx'[p(x' ) - Pol, "0
(2.24)
where we have speCifically neglected any phases coming from fat' dt'. 13 Equation (2.24) is just the WKB result, minus a phase of 1T/4. It appears reasonable to assume that the neglected coritribution to Eq. (2.24) due to at is something like 1T/4. 13
The two techniques illustrated above, namely Fourier time transformation and asymptotic state projection, are important means of extracting state-speCific information from the wave packet trajectories. Both techniques are founded on the quantum principle of superposition. That is to say, the wave packets are a linear superposition of eigenstates, and the eigenstates are superpositions oj wave packets. Since our technique of expanding the potential about the center of the wave packet depends upon having well-localized wave packets, actual dynamical calculations are always carried out with such states. The superposition principle then tells us what linear combinations and projections of the individual "quantum trajectories" to take to obtain the desired state-specific results. We shall have more to say on this point in subsequent sections.
We conclude this section with a concrete and useful example of the projection-Fourier transform techniques, applied to the harmonic oscillator. We choose as an
FIG. 1. (a) A Gaussian wave packet at its turning point in a !>Be-dimension potential. (b) Fourier time transform of the wave packet appearing in (a).
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1548 Eric J. Heller: Time-dependent approach to semiclassical dynamics
initial value
ao=imw/2,
and as previously noted, for this choice we have at = ao [see Eqs. (2.l1c) or (2.13)]. The remaining parameters characterizing the Gaussian wave packet are (using Y as the coordinate)
We wish to extract an eigenstate from Eq. (2. 26a). In accordance with our previous discussion on obtaining eigenstates and from Eq. (2. 26c), we can write
w fbi", 1/!"(Y) = 21TC
n(0)J
o exp[i(n +t)wt]1/!(y, t)dt . (2.27)
Having used a Fourier time transform to obtain 1/!"(y), we now employ a projection onto 1/!" to obtain c"(O):
(2.28)
The overlap may be obtained easily by appealing to the theory of the coherent state, 11 which tells us
1/!(y, t) = exp(-I f30 12/2 - iwt/2) t f3;/FnT x My) , (2.29)
n=O
where
{it = (mw/21i)1/2(yo + ipo/mw)e- iwt ,
thus
c"(0)=exp(-1f30 12/ 2)f3;;;rnr . (2.30)
The desired representation of rfJ"(y) in terms of a superposition of wave packet states is obtained by inserting Eq. (2.30) into Eq. (2.27), giving
Equation (2.31) is an integral representation of the oscillator functions which may be apprOXimated by an N-point quadrature in t such that the resulting inexact wavefunction has no components in the nearest N - 1
neighboring oscillator states on either side. This is shown in the Appendix, and used in the numerical work of Sec. V.
In the oscillator case, the Fourier transform of the wave packet states results in an exact representation of the eigenfunctions. This is in contrast to the WKB wavefunctions for the harmonic oscillator, which are inexact. This emphasizes again the difference between a locally quadratic expansion of the potential and an expanSion in 1i.
III. SYSTEMS OF SEVERAL VARIABLES
A. The wavefunction and correlation terms
The techniques and results of Sec. n generalize readily to nonseparable dynamical systems of higher dimension. We illustrate the method by working out a two-dimensional system which might, for example, correspond to atom-diatom collinear inelastic and reactive scattering on a potential surface V(x, y).
In direct analogy with Sec. n, we will expand the potential in a Taylor series about the center of the twodimensional wave packet
and for the purpose of propagating a given wave packet, we substitute Vt for V in the Hamiltonian. The only new twist is the correlation term VXy(x- xt)(y - Yt) which makes the Hamiltonian nonseparable, since in general VXy is time dependent. This term implies that we cannot expect to obtain an exact solution even to the approximate, time-dependent Hamiltonian H t = Ho + Vt if the wavefunction is written in the form 1/!(x, t) x cfJ(y, t) since this form contains no x, Y correlation. However, this situation is easily remedied by inclusion of the proper correlation term in the wavefunction, which takes the form
We have retained the subscript t on the quantities x" Yt to distinguish them from the variables x, y; otherwise, we drop the explicit time dependence, it being understood in ax, a y, Px, Py, Y, and A.
The role played by the x, Y correlation in the wavefunction is an interesting one. While it is true that the separable form 1/!(x, t) x cfJ(y, t) contains no x, Y correlation, this is not true of the linear superpositions of such functions which are required to approximate quantum
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Eric J. Heller: Time-dependent approach to semiclassical dynamics 1549
eigenstates. Moreover, in the semiclassical limit, the linear terms in Eq. (3.1) will outweigh the quadratic terms for narrow wave packets. Thus the separable Hartree-like approximation may be a viable semiclassical approximation in some situations. However, explicit inclusion of x, y correlation is desirable if it can be done, since then the effective Hamiltonian Ht is closer to H. From another point of view, A is on equal footing with a" and a y; a simple rotation of coordinates shows this.
B. Equations of motion
To determine equations of motion for the parameters appearing in Eq. (3.4), we proceed as before, substituting Eq. (3.4) into the Schrodinger equation. Using Ht '" Ho + Vt in place of H, we find that the resulting equations can be satisfied if the parameters obey
• aH x-t - ap,,'
• aH P,,=- aXt '
• aH Yt = apy ,
· (a ay) A=-2 .::.L+_ A-V"y, m" my
(3.5a)
(3.5b)
(3.5c)
(3.5d)
(3.5e)
(3. 5f)
Similar equations can be derived for systems of higher dimensionality. The last three terms in the expression (3. 5f) for Y give rise again to the action integral along the classical path.
IV. APPLICATION TO COLLISIONS: OBTAINING THE SMATRIX
A. General considerations
It stands to reason that it will usually not be possible to extract quantum state specific information from a single quantum wave packet trajectory. A de localized quantum state contains information about the dynamics over its entire extent; a single trajectory may only sample part of this dynamics. Therefore, only by performing several trajectory calculations and summing the trajectories in a coherent fashion can we hope to extract an S matrix for a system involv.ing two or more coordinates. For two or more coordinates, there may be many dynamically distinct trajectories lying on the same energy surface, each corresponding to a different set of classical initial conditions.
The superposition principle guides us in choosing an optimum linear combination of trajectories to correspond to specific quantum transitions. However, as will be seen below, there is some flexibility in the way in which the individual trajectories are chosen. In Secs. IV. B and IV. C, we treat two methods which suggest themselves.
B. Controlled incoming state trajectories
Long before a collision has occurred between two dynamical systems, the interaction between them vanishes, and it is permissible to write the total wavefunction in the separable form
1Ji1n = ¢(Y, t)lf;(x, t) , (4.1)
where now Y is a collective coordinate labeling the constituents of the (initial) target system. For example, 1> might correspond to a normalized quantum eigenstate ¢no of the target, and If; an incoming Gaussian wave packet. In this case, 1Ji1n can be written
1Ji1n,nO = ¢nO(Y) exp( - i/IiEnot)j(kno)1/2
x J"' dkA(k)exp( - ikx- ik2/2mxt) , (4.2) ."'
with A(k) chosen to give the proper Gaussian form. Then the outgoing state lJiout can be written
lJiout '= -~ 1: dkA(k)s"on' (Ek,no)/(kn• )1/2
X ¢n' (y) exp(ik",x - i/ IiEk,not), (4.3)
where Ek,nO '= Eno + k2/2m", kn' = [2mx(Eno - En' + k2/2m,,)]1/2, and S"on' (Ek,nO) is the usual scattering S matrix. Thus we can find S"On' as
S"o", (Eko,nO) = [21TA(kO)].le"~EkO.nOt kO /(kn• )1/2
(4.4)
where k~,=[2m"(Eno-E,,,+ko2/2m,,)]1/2. However, 1Ji1n
of Eq. (4.1) is not, in general, an acceptable incoming state if ¢"o is an eigenstate, since then ¢nO is not a wave packet. If we know a wave packet representation for the eigenstate ¢"o, such as has already been derived for the harmonic oscillator in Sec. ll. C (see also the Appendix), we can write
== [~ ano,g,(y, t)] If;(x, t) , (4.5)
where the sum in the brackets in Eq. (4.5) is an approximation to ¢"o exp( - i/ IiEnot) and each g, is a Gaussian wave packet. Then each of the fully Gaussian states 1Ji~!) is propagated individually according to our prescriptions, and we have
(4.6)
Applying this latter form for lJiout in Eq. (4.4) gives us a practical scheme for computing S matrix elements. Actually, it is possible to be more general than this; each incoming wave packet 1Ji~!1 may take the form
1Ji~!1 =g,(y, t)If;,(x, t) , (4.7)
and a very similar procedure follows for finding the desired S-matrix elements.
The sense in which several trajectories are needed is
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1550 Eric J. Heller: Time-dependent approach to semiclassical dynamics
now clear from Eqs. (4.5) and (4.6). It is also clear that a single quantum trajectory -li'lu~out can be used in the computation of many S-matrix elements. For example, we obtain a whole column of S, namely SnOn" n' = 1,2, ••. from -li'tn,.nO in Eq. (4.5). Furthermore, we may be able to generate a different incoming state,
from the same set of initial states, yielding more S-matrix columns.
C. Controlled collisional state trajectories
Naturally it is desirable to choose the initial parameters of the wave packet so that it remains as compact as possible for the duration of the interaction. Although simple procedures may be devised for choosing a good set of initial parameters (in particular, the "spread" parameter a), as is done in the numerical application in Sec. V, an alternative procedure is to choose the quantum parameters to have some desirable values during the most arduous part of the trajectory. Then this "collisionally controlled" state is propagated forward and backward in time:
(4.8)
>lt~!} will no longer be separable in x and y, but the reversibility of the equations of motion tells us that starting with -li'~~) of Eq. (4. 8), we have
(4.9)
The different quantum trajectories (i) may all lie on a given (classical) energy surface, or they may be further restricted by requiring the target to have a certain clasSical energy initially.
The most rigorous treatments of scattering theory14 begin with time-dependent wave packets, since these are properly normed vectors in Hilbert space, while the energy eigenstates for a scattering system are not. The transition from time-dependent to time-independent scattering theory has typically been a device used to justify the time-independent form of the theory. Here, we can use the same formal steps to provide another viewpoint on extracting eigenstate information from our wave packet trajectories.
We further specify our asymptotic wave packets by defining -li'1!} to be equal to the true wave packet for the system fot large negative t (long before collision), and similarly for -li'~!~ for large positive t. We require that both functions propagate according to the free Hamiltonian Ho at all times. Then, for any time t, the S operator performs the in-out transformation14
:
(4.10)
S is time independent. Thus a Fourier transform of Eq. (4.10) gives
(4.11a)
or
where the sums in Eq. (4.l1b) are over all degenerate arrangements of energy among the fragments, 1. e. , over all open channels. Defining
27T /i2
Snn' o(E _ Eo) = <¢neik~X /vikg I sl ¢"..e~i~'x /~'I , ) , m
x (4.12)
where E = En + /i2 lf,,/ (2mx), etc., we obtain, by projection onto the nth channel in Eq. (4.llb),
(4.13)
or
(4.14)
In practice, the A's and B's are determined by projecting the wave packet before and after the collision onto the known asymptotic eigenfunctions
¢n(y)e-ik"x /fli;. ,
etc. Selecting as many distinct trajectories (or independent linear combinations of trajectories) as there are open channels, we determine the whole S matrix at once from Eq. (4.14)
V. NUMERICAL APPLICATION: COLLINEAR INELASTIC COLLISION
A. The model and details of the calculation
As the Simplest nontrivial test case, we deal with collinear atom-diatomic collisions in which reaction is not allowed. The familiar model investigated quantum mechanically by Secrest and Johnson15 is used, in which the diatom is a harmonic oscillation which interacts with the projectile atom via an exponential repulsion. Transforming coordinate systems and using convenient units allows us to write the Hamiltonian as
Corresponding to an He+ H2 colliSion, we have a= O. 3 and m" = 2/3.
The over-all scheme described in the previous sections can easily be applied to this model Hamiltonian. The target oscillator eigenstates, which have the exact wave packet representation, Eq. (2.31), are approximated as finite wave packet sums. As shown in the Appendix, an N term sum can be chosen so that the approximate eigenstate has no components in any of the N-1 eigenstates on either side of the chosen state. We use the incoming state representation. The N-term wave packet sum corresponding to the desired initial quantum state of the oscillator is equivalent to N different initial phase angles of the classical oscillator spaced evenly on (0,211'). In practice, N = 4 or 8 is more than sufficient for the work reported here.
An incoming wave packet state, where both the target and free particle are represented as Gaussians moving on asymptotic potential surfaces, has components in all
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Eric J. Heller: Time-dependent approach to semiclassical dynamics 1551
of the asymptotic quantum eigenstates of the system, including those with energies very different from the classical energy of the Gaussians. Thus, conversely, it is possible to add coherently a series of such wave packet states, all of which have classical energy E, to give a quantum incoming state of energy E' which may be much different from E. However, since we have only an approximate scheme for propagating the wave packets, it would seem best to use quantum trajectories with classical energies close to the energy of the quantum state being investigated. If this is done, we get the additional benefit that fewer quantum trajectories will be needed to define a specific eigenstate, since eigenstates far removed in energy will have very small amplitude in the wave packets and need not be specifically eliminated by the coherent sum of quantum wave packet trajectories.
Thus, in our calculations we require the oscillator to initially have classical energy no or no+ t, and the free particle to have classical energy Eo, if it desired to describe an incoming quantum state with the oscillator in the noth quantum state and the projectile in the state exp( - iltx) /ffo, ko = ..f2Eo • This means that the classical energy will be within 1/2 oscillator quantum of the eigenstate energy. For highly excited states of the OSCillator, it certainly will not be necessary to match the classical initial conditions to this degree to the quantum states. However, we have not yet explored how far "off shell" a quantum trajectory may be before it gives poor information about a "distanf'eigenstate.
We also have the freedom to choose a,. and ay initially. The latter is always taken to be 1/2, which is the value which would remain unchanged in the absence of a collision. The parameter a,. time develops, in the absence of collisions, as
so we can choose ao and T to keep the wave packet as small as possible during the collision. We choose ao = (p ~2 /M, where M = 30, 40, or 100, as noted in the tables, where p2 is the initial value of p,.. T was chosen from among several sample T's to give a minimum error estimate due to the cubiC part of the potential (only Vxu was taken into account). The main idea is to arrange for the wave packet to "collect itself" just as it collides strongly; the results are not critically dependent on T in any case.
After the wave packet emerges again into the asymptotic region, we perform the overlaps
do') =fdXdY[ e-y2/2Hn(Y)] e- il•g.,. /~ (2"n! fi)1/2 n
where the parameters <l:., etc. all refer to the ith trajectory. These are readily done analytically by first performing the Gaussian (dx) integration and then evaluating the (dy) integral of a Hermite polynomial with a shifted Gaussian. Such amplitudes are added coherently (with proper coefficients Anol for the ith trajectory). Finally, we obtain the inelastic transition probability
Pnon:
Pnon = I ~Antl d/) \ Z (5.3)
The first order equations (3.5) are solved with a fourth order Runge-Kutta routine; the time step size is typically O. 03.
B. Results
In Fig. 2 we present results for the He+ Hz system for total energies of 8 and 10, comparing with Secrest and Johnson.15 Table I gives numerical values for Etot = 3, 4, 5, 6, 8, and 10. The classical oscillator energy initially was n for trajectories used to compute Pn - n,.
Table II shows similar results, which may be compared with Table I, for the case in which n+ t was the classical initial oscillator energy.
The individual wave packets were found to remain unit normed to eight or nine Significant figures. However, since these wave packets are added coherently, and since each different quantum trajectory is acted upon by a different effective Hamiltonian, there is no theorem guaranteeing unitarity. It is gratifying that the individual Pnn' probabilities add up to unity to within 1% in almost all cases (see Table I).
Since in these calculations Pnn, is computed using different trajectories than are used for Pn'n, they need not agree, and this is seen in the tables.
A few pOints about the trajectories are worth noting. A typical maximum value during the trajectory for Im(a,.), which is the parameter responsible for most of the spread of the wave packet, was O. 2 or O. 3. The typical behavior of Im(a,.) is to increase to a maximum, decrease, and then increase again before finally declining as the particles separated.
The results are reasonably stable against variation in the choice of incoming parameters. A very poor choice
PO,n 0.2 - Secrest ond Po n 0.2 Johnson' '
E = 10 H2+He -- - Present 0.4 f\~~!-H~resent 0.4
'" 0.0 LL---1---1-----'-----'_'----'----'-
n- n-
FIG. 2. The "exact" transition probabilities of Secrest and Johnson (Ref. 15) are compared to the present quantum trajectory calculations, for E = 8 neft) and 10 (right), corresponding to an He + ~ collision. Table I contains numerical values for these and other energies.
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1552 Eric J. Heller: Time-dependent approach to semiclassical dynamics
TABLE I. Transition probabilities Pi~1 for He + H2, E tot = 3, 4, 5, 6, 8, 10. "QT" refers to the present result, "SJ" to Ref. 15. The values in brackets are inferred from the inelastic results of Ref. 15. Shown also are the sums L;"". =L:n'P""" For energies of 3, 6, and 10, an asterisk (*) indicates a transition that is classically forbidden [Miller, Ref. 4(b), while (**) indicates that a tral).sition and all those below it in the table, for fixed initial state, are likewise classically forbidden. For E tot = 3 and 4, Ci 0
= (PZ)2/30; for E tot = 5, 6, 8, 10, CiO = (p2)2/40.
can result in nonsense, otherwise the results retain their character almost to eyeball accuracy in Fig. 2, for example, although very small transition probabilities may change more on a percent basis. There is, however, a definite tendency for less extended wave packets to give over-all better agreement with the exact values, as would be expected.
We have reason to suspect that use of the controlled collisional state representation will provide even better results than have been obtained so far, because of the greater flexibility in chOOSing well-localized packets.
Finally, we present in Table m results for a different set of parameters, namely a= 0.114 and M" = 0.5. This system was chosen because the exact transition probabilities computed by Secrest and Johnson15 are extremely small. Nonetheless, the accuracy of the quantum trajectory results is still good, although not surprisingly the results were more sensitive to numerical errors (the t-step size was reduced to 0.015). The good agreement is partly explained by the fact that the wavepackets were very well behaved on this extremely smooth potential surface.
VI. EXTENSIONS AND FURTHER WORK
In previous sections we found that it was possible to include all quadratic terms in the Taylor expansion of the potential about the center of the wave packet and still obtain an exact solution, for each Gaussian wave packet,
TABLE n. Results for He + H2 system with E ... initially i + i, where i is initial oscillator state. 010= (P~)/30.
Etot,L
5 0 5 0 5 0 5 0 5 0
~On 5 1 5 1 5 1 5 1 5 1
LIn
f
0 1 2 3 4
0 1 2 3 4
pW
0.735 0.236 0.177 0.192 (-3) 0.145 (-6) 0.989
0.276 0.581 0.144 0.651 (-2) 0.954 (-4) 1.007
[0.733) 0.252 0.0152
1.000
0.252 [0.563) 0.182 0.331 (-2)
1. 000
TABLE III. Transition probabilities for Secrest and Johnson system numbe'r n, M,,= 0.5, 01 = 0.114. a
&six quantum trajectories were used at each of the two energies, E tot =3.8 and 4.4. 010= (p~)2/100. Eoe. is initially (Hi).
to the effective time-dependent Hamiltonian. One may ask if it is possible to include cubic and higher derivatives of the potential in the effective Hamiltonian. The answer is yes, if we are willing to give up finding an exact solution to the approximate, time-dependent Hamiltonian.
The procedure which generalizes the Gaussian approach used up to now is to write the wavefunction in the form
l/J(x, t) = N(x, t) exp [i/liQ(x, t)] , (6.1)
where N(x, t) is a polynomial of degree N, of the form
N(x, t) = l+n, x+n2x2+... • (6.2)
Q(t) is again a quadratic. Substitution of the form (6. 1) into the Schrodinger equation, followed by identification of coefficients of like powers of x, results in the problem that the term
V(x, t)N(x, t) exp [i/liQ(x, t)] , (6.3)
where V is the time-dependent effective potential, generates powers of x greater than N + 2, if V is a polynomial of degree greater than 2. But N +2 is the limit on powers of x for which unknowns are available. The only alternative is to truncate Eq. (6. 3) to terms of power N+2 or lower. This procedure should result in an improvement in accuracy over the purely Gaussian form, but some SimpliCity is lost.
If for some reason (e. g., tunneling) a Gaussian wave packet should break up into two or more packets, not even Eq. (6. 1) will help. We might pick a wavefunction which is a sum of several Gaussians and see what happens. We are straying into the subject matter of a following paper, 16 where we develop the notion that much more general guesses for the time-dependent form of the wavefunction may prove useful in semiclassical dynamics. We also break away from the Taylor expansion of the potential and consider a more powerful time-dependent variational approach.
However, much remains to be done with the strictly Gaussian wave packet methods developed here. For example, we don't yet know the optimum way of extracting bound states for nonharmonic potentials. And, as mentioned before, barrier tunneling seems to be a problem requiring further attention. Wave packets in nonCartesian coordinates need to be investigated. And potential applications to a variety of physical systems lie ahead.
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1554 Eric J. Heller: Time-dependent approach to semiclassical dynamics
CONCLUSION
In spite of the intuitive appeal of an explicitly timedependent, particle-like description of semiclassical dynamics, efforts along these lines have been rather sparce. In quantum mechanics, it is usually much easier to directly construct eigenstates than to propagate time-dependent wavefunctions. However, the explicitly time-dependent procedures successfully employed in classical dynamics suggest that similar techniques may become practical in the semiclassical limit. In this paper we have discussed such a technique, which does indeed seem to benefit from this time dependent view of the correspondence prinCiple. The first order equations of motion derived previously are simply Hamilton's equations for the classical variables, while the quantum variables obey formally similar first order quantum equations. Given a set of initial conditions, the whole system of variables simply executes appropriate classical or quantum trajectories.
Several of the previous attempts to exploit or examine the time-dependent wave packet version of quantum mechanics are especially worthy of mention in the context of this paper. McCullough and Wyatt17 have undertaken a careful study of the explicit quantum dynamiCS of wave packets on a collinear H + H2 surface. Their wave packets were not chosen to be optimally "small" during the collison, but their results give several insights into the dynamics, not the least of which is the appearance of vortices in the wavefunctions. Recently, Hirshfelder, Goebel, and Bruch18 have noted that such vortices must form about nodes in the wavefunction.
Perhaps closest in spirit to the work reported in the preceeding sections is the work of Lebedeff. 19 He also considered the propagation of Gaussian wave packets. However, his formalism prevented the inclusion of any terms in the potential higher than linear in the coordinates. ThUS, the resulting wave packets are uncorrelated (missing any analogue of the A term in Eq. (3. 5), a fact which would lead to serious error in the He + H2 calculations of Sec. V. A more serious error arises because even a linear harmonic oscillator is propagated incorrectly. Nonetheless, Lebedeff's work represents a significant step in the direction we have chosen in this paper.
Recently Hirschfelder, Palke, and Christoph20 have studied time-independent wave packets (in the form of focused beams) in a study motivated by the hydrodynamic form of the Schrooinger equation. Their study emphasizes nicely the differences between a wave packet description and the stationary phase, 1£- 0 version of quantum mechanics. For example, the stationary phase analogue of their wave packet does not show any diffraction, nor does it describe the Goos-Hanchen shift properly. Wave packets have the ability to overcome many of the "caustic" difficulties of the stationary phase approximation.
It is worthwhile to point out some additional comparisons between a wave packet approach and a stationary phase approach. To a large extent both techniques are "classical mechanics plus quantum superposition." Our
methods require that we solve for the classical and quantum trajectories. add coherently the results of several such trajectories, and project the results onto the desired energy surface. Since each quantum trajectory contains dynamical information about a region of phase space, each wave packet has some information to contribute regarding a specifiC quantum transition. ThUS, the classical action of the wave packets need not conform to integer values, for example, as in the classical S-matrix theory. The requisite trajectories for a particular calculation are determined at the outset, not by a search procedure. However, on the other side of the coin, it may be necessary to run several quantum trajectories corresponding to a single classical trajectory in order to find initial conditions which keep the wave packets well localized during collision.
The fact that the approach presented in this paper is exact for harmonic potentials is encouraging for applications to molecular dynamiCS, where the oscillator description has long been a good starting point. Thus, the method may be useful in the study of intra- and intermolecular energy exchange and unimolecular decomposition. Since wavefunctions are obtainable from the wave packet trajectories, photodissociation cross sections can be calculated. Because the wavefunctions are represented as sums of Gaussians (see the Appendix, for example), the requisite integrals are greatly simplified.
Other important processes which have a direct classical analogue will be amenable to the quantum trajectory approach. Still others, which have significant nonclassical aspects (such as curve crossing and reactive tunneling) require generalizations of the present work that are currently under investigation. A more general and powerful variational approach to time dependent semiclassical dynamiCS is discussed in a following paper. 16
ACKNOWLEDGMENTS
The author is indebted to Professor Stuart Rice for support, encouragement, and helpful discussions during the course of this work. Numerous conversations with Professor Karl Freed and Dr. Sture Nordholm were also of great benefit. This work was partially supported by Air Force contract AFOSR F44620-73-C-0005.
APPENDIX: APPROXIMATE REPRESENTATION OF OSCILLATOR EIGENSTATES AS DISCRETE SUMS OF GAUSSIAN WAVE PACKETS
The time-dependent, oscillating Gaussian wave packet, Eq. (2.26a), has a representation in terms of the unit normalized, time independent oscillator eigenfunctions of the form
l/i(y, t) a: L: exp[ - i(n + ~ )wt]tl~/rnT x IjJ" • (Ai) n
The l/i.' s are the eigenstates we seek to represent as finite wave packet sums. It is clear from Eq. (Ai) that
ih'W
l/in a: exp[i(n + ~ )wt]l/i(y, t)dt o
(A2)
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Eric J. Heller: Time-dependent approach to semiclassical dynamics 1555
ib'W CTi
ex L exp[i(n - n')wt],98' / yn ll/Jn' • "' 0
(A3)
Replacing the integral in Eq. (A3) with a discrete quadrature in t, we choose tp = 21rp/(Nw) and we have
where the tilde reminds us that we now have an approximate lJi". Since
N
..!.. L exp[i(n - n')21rp/N] = ('In.", + c5 N•1" .... 'I+ c52N • 1" .... '1 + ••• , N p=l (A 5)
we have from Eq. (A4) that
13" an.N
1- > ~ 1-'0 (l/J". l/J" ex If c5n", + -~N f ON.J"-n'I+ •••
'In! ·V\n"'l~/!
(A6)
ThUS, for example, ~"has no components in the nearest (N - 1) oscillator states on either side of n. The exact formula for ~n is just Eq. (2.31), with w/2rr f~~/Wdt replaced by I/N'L,:=l; tp = 2rrp/(Nw), i. e.,
+i P.o(Y - Yp) + 2~ (PPYP - PoYo) + in x 2rrp/NJ ' (A 7)
where
YP = Yo cos (21fp/N) + (pol mw) sin(2rrp/N), etc.
Equation (A 7) is simply a sum of Gaussian wavefunctions. Note that as long as Eo=pV2m+tmw2y~>O, any 1/1" can be approximated using (A 7). However, the error terms for ~" will be smallest [see Eq. (A6)] if ,90' /.fil'f is a maximum for n' = n. [As before, we define ,90 as ,90 = (mw/21f)1/2(yo + iPoi mw ).] This maximum occurs if Eo "'(n+t)w.
Finally, we note that Eq. (A7) may be of use in other circumstances. For example, if we wish to perform the integrals
(A8a)
or
(A8b)
where 1/1".' s are oscillator eigenfunctions, we can substitute Eq. (A 7) into Eqs. (A8). The 1/Jn's are then replaced by Gaussians, and the new integrals will usually be much easier to perform analytically.
We call this technique a nonlocal Gaussian quadrature. It has some features analogous to the usual theory of Gaussian quadrature. 21 For example, suppose Eq. (A8a) is of the form
where H"(y) is the nth Hermite polynomial and p(y) is a polynomial of degree 2n or less. Then I" is given exactly if we substitute $" for 1/J" in Eq. (A9) with N in Eq. (A7) at least as large as n + 1.
lW. H. Miller. J. Chern. Phys. 54. 5386 (1971). 2p. A. M. Dirac. Quantum Mechanics (Oxford U. P .• London,
1958). 3R. P. Feynman and A. R. Hibbs, Quantum Mechanics and
Path Integrals (McGraw-Hill. New York. 1965). 4(a) W. H. Miller, J. Chern. Phys. 53. 1949 (1970); (b) 53.
3578 (1970); (c) 54. 5386; (d) 55. 3150 (1971). For a recent review and additional references, see W. H. Miller, Advances in Chemical Physics (Wiley. New York. 1974). Vol, XXV.
5(a) R. A. Marcus. Chern. Phys. Lett. 7, 525 (1970); (b) J. Chern. Phys. 54, 3965 (1971); (C) J. N. L. Connor and R. A. Marcus, J. Chern. Phys. 55, 5636 (1971); (d) w. H. Wong and R. A. Marcus. J. Chern. Phys. 55, 5663 (1971); (e) R. A. Marcus. J. Chern. Phys. 56, 311 (1972); (f) 56, 3548 (1972); (g) J. Stine and R. A. Marcus. Chern. Phys. Lett. 15, 536 (1972); (h) R. A. Marcus. J. Chern. Phys. 57, 4903 (1972); (i) D. E. Fitz and R. A. Marcus, J. Chern. Phys. 59, 4380 (1973).
6(a) M. C. Gutzwiller, J. Math. Phys. 8, 1979 (1967); (b) 10, 1004 (1969); (f) 11, 1791 (1970); (d) 12, 343 (1971).
7B. C. EU, J. Chern. Phys. 57, 2531 (1972). 8See , for example. K. Gottfried, Quantum Mechanics (Benja
min, New York, 1966). 90f course, if n is in some sense small, then d will generally
tend to remain small, so that in the presence of nonharmonic potentials, small·n does playa role in keeping the Ehrenfest error term small.
lOSee, for example, R. P. Feynman and A. R. Hibbs, Ref. 3. I1W. H. Louisell, Quantum Stntistical Properties of Radiation
(Wiley, New York, 1973). 12H• Goldstein, Classical Mechanics (Addison-Wesley, Reading,
MA, 1950). 13 As noted earlier, CI t can add a pure phase factor to the wave
function. Presumably this is a correction to the remaining phase. Its effect has been neglected for this discussion; in any explicit calculation involving Eqs. (2.11), the phase of CIt is of course included.
USee, for example. R. G. Newton, SCattering Theory of Waves and Particles (McGraw-Hill, New York. 1966).
15D. Secrest and B. R. Johnson, J. Chern. Phys. 45, 4556 (1966).
16E. J. Heller (in preparation). 17(a) E. A. McCullough and R. E. Wyatt, J. Chern. Phys. 54,
3578 (1971); (b) 55, 3592 (1971). 18J . O. Hirschfelder, C. J. Goebel, and L. W. Bruch, Wiscon
sin Theoretical Chemistry Institute Report WIS-TCl-515, 1974.
19(a) S. A. Lebedeff, Phys. Rev. 165, 1399 (1968); (b) Phys. Rev. D 1, 1583 (1970). The author wishes to thank Professor R. E. Wyatt for bringing this work to his attention.
20J. O. Hirschfelder, A. C. Christoph. and W. E. Palke, Wisconsin Theoretical Chemistry Institute Report WIS-TC-514, 1974.
2l See, for example, E. Isaacson and H. B. Keller, AnalySis of Numerical Methods (Wiley, New York. 1966).
J. Chern. Phys., Vol. 62, No.4. 15 February 1975
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